Fast spatial Gaussian process maximum likelihood estimation via skeletonization factorizations

TL;DR

This paper introduces a fast computational framework for maximum likelihood estimation of Gaussian processes in two dimensions, significantly reducing the time complexity for large datasets by leveraging skeletonization and matrix peeling techniques.

Contribution

It develops a novel method combining skeletonization and matrix peeling to efficiently evaluate likelihoods and gradients for Gaussian process parameter estimation in 2D.

Findings

Achieves $ ilde O(n^{3/2})$ time complexity for likelihood evaluation.

Enables scalable Gaussian process parameter fitting for large datasets.

Provides a framework compatible with standard optimization routines.

Abstract

Maximum likelihood estimation for parameter-fitting given observations from a Gaussian process in space is a computationally-demanding task that restricts the use of such methods to moderately-sized datasets. We present a framework for unstructured observations in two spatial dimensions that allows for evaluation of the log-likelihood and its gradient (i.e., the score equations) in $\tilde O(n^{3/2})$ time under certain assumptions, where $n$ is the number of observations. Our method relies on the skeletonization procedure described by Martinsson & Rokhlin in the form of the recursive skeletonization factorization of Ho & Ying. Combining this with an adaptation of the matrix peeling algorithm of Lin et al. for constructing $\mathcal{H}$-matrix representations of black-box operators, we obtain a framework that can be used in the context of any first-order optimization routine to quickly and accurately compute maximum-likelihood estimates.